ESTADISTICA PARA EL ANALISIS POLITICO II
- Profesor: Dr. José Manuel Magallanes, Ph.D.
- Profesor del Departamento de Ciencias Sociales, Sección de Ciencia Política y Gobierno.
- Oficina 105 - Edificio CISEPA / ECONOMIA / CCSS
- Telefono: (51) 1 - 6262000 anexo 4302
- Correo Electrónico: jmagallanes@pucp.edu.pe
Análisis Factorial I: Exploración
Hacemos analisis factorial para reducir las variables en otras variables resumen. Mientras la clusterización agrupaba filas, la factorización agrupa columnas. Pero, al igual que en clusterización, queremos saber si las nuevas variables tienen un nombre, al cual se le denomina técnicamente variable latente. En esta sesión exploraremos la data a ver qué emerge.
Preparación de Datos:
Para esta sesión trabajaremos con la data de estos links:
library(htmltab)
# links
happyL=c("https://en.wikipedia.org/wiki/World_Happiness_Report",
'//*[@id="mw-content-text"]/div/table/tbody')
demoL=c("https://en.wikipedia.org/wiki/Democracy_Index",
'//*[@id="mw-content-text"]/div/table[2]/tbody')
# carga
happy = htmltab(doc = happyL[1],which = happyL[2],encoding = "UTF-8")
demo = htmltab(doc = demoL[1], which = demoL[2], encoding = "UTF-8")
# limpieza
happy[,]=lapply(happy[,], trimws,whitespace = "[\\h\\v]") # no blanks
demo[,]=lapply(demo[,], trimws,whitespace = "[\\h\\v]") # no blanks
library(stringr) # nombres simples
names(happy)=str_split(names(happy)," ",simplify = T)[,1]
names(demo)=str_split(names(demo)," ",simplify = T)[,1]
## Formateo
# Eliminemos columnas que no usaremos esta vez:
happy$Overall=NULL
demo[,c(1,9,10)]=NULL
# También debemos tener nombres diferentes en los scores antes del merge:
names(happy)[names(happy)=="Score"]="ScoreHappy"
names(demo)[names(demo)=="Score"]="ScoreDemo"
# Tipo de variables:
## En demo:
demo[,-c(1)]=lapply(demo[,-c(1)],as.numeric)
# En happy:
happy[,-c(1)]=lapply(happy[,-c(1)],as.numeric)
# sin perdidos:
happy=na.omit(happy)
demo=na.omit(demo)Presta atención al merge. Usualmente hacemos merge por default y luego perdemos varias filas:
nrow(merge(happy,demo))## [1] 145Hagamos un nuevo merge, donde nos quedemos con TODOS los paises que no estab en uno u otro data frame:
HappyDemo=merge(happy,demo,all.x=T, all.y=T)Esta vez HappyDemo tiene varios paises de más, pero con valores perdidos y nombres que no pudieron coincidir. Veamos:
# formateando a
# HappyDemo[!complete.cases(HappyDemo),]
library(knitr)
library(kableExtra)
kable(HappyDemo[!complete.cases(HappyDemo),],type='html')%>%
kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"),
font_size = 10)| Country | ScoreHappy | GDP | Social | Healthy | Freedom | Generosity | Perceptions | ScoreDemo | Electoral | Functioning | Politicalparticipation | Politicalculture | Civilliberties | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 14 | Belize | 5.956 | 0.807 | 1.101 | 0.474 | 0.593 | 0.183 | 0.089 | NA | NA | NA | NA | NA | NA |
| 27 | Cape Verde | NA | NA | NA | NA | NA | NA | NA | 7.88 | 9.17 | 7.86 | 6.67 | 6.88 | 8.82 |
| 33 | Comoros | NA | NA | NA | NA | NA | NA | NA | 3.71 | 4.33 | 2.21 | 4.44 | 3.75 | 3.82 |
| 34 | Congo (Brazzaville) | 4.559 | 0.682 | 0.811 | 0.343 | 0.514 | 0.091 | 0.077 | NA | NA | NA | NA | NA | NA |
| 35 | Congo (Kinshasa) | 4.245 | 0.069 | 1.136 | 0.204 | 0.312 | 0.197 | 0.052 | NA | NA | NA | NA | NA | NA |
| 38 | Cuba | NA | NA | NA | NA | NA | NA | NA | 3.00 | 1.08 | 3.57 | 3.33 | 4.38 | 2.65 |
| 41 | Democratic Republic of the Congo | NA | NA | NA | NA | NA | NA | NA | 1.49 | 0.50 | 0.71 | 2.22 | 3.13 | 0.88 |
| 43 | Djibouti | NA | NA | NA | NA | NA | NA | NA | 2.87 | 0.42 | 1.79 | 3.89 | 5.63 | 2.65 |
| 48 | Equatorial Guinea | NA | NA | NA | NA | NA | NA | NA | 1.92 | 0.00 | 0.43 | 3.33 | 4.38 | 1.47 |
| 49 | Eritrea | NA | NA | NA | NA | NA | NA | NA | 2.37 | 0.00 | 2.14 | 1.67 | 6.88 | 1.18 |
| 51 | Eswatini | NA | NA | NA | NA | NA | NA | NA | 3.03 | 0.92 | 2.86 | 2.22 | 5.63 | 3.53 |
| 53 | Fiji | NA | NA | NA | NA | NA | NA | NA | 5.85 | 6.58 | 5.36 | 6.11 | 5.63 | 5.59 |
| 57 | Gambia | NA | NA | NA | NA | NA | NA | NA | 4.31 | 4.48 | 4.29 | 3.33 | 5.63 | 3.82 |
| 64 | Guinea-Bissau | NA | NA | NA | NA | NA | NA | NA | 1.98 | 1.67 | 0.00 | 2.78 | 3.13 | 2.35 |
| 65 | Guyana | NA | NA | NA | NA | NA | NA | NA | 6.67 | 9.17 | 5.71 | 6.11 | 5.00 | 7.35 |
| 84 | Kosovo | 5.662 | 0.855 | 1.230 | 0.578 | 0.448 | 0.274 | 0.023 | NA | NA | NA | NA | NA | NA |
| 95 | Macedonia | 5.185 | 0.959 | 1.239 | 0.691 | 0.394 | 0.173 | 0.052 | NA | NA | NA | NA | NA | NA |
| 117 | North Korea | NA | NA | NA | NA | NA | NA | NA | 1.08 | 0.00 | 2.50 | 1.67 | 1.25 | 0.00 |
| 118 | North Macedonia | NA | NA | NA | NA | NA | NA | NA | 5.87 | 6.50 | 5.36 | 6.67 | 3.75 | 7.06 |
| 119 | Northern Cyprus | 5.835 | 1.229 | 1.211 | 0.909 | 0.495 | 0.179 | 0.154 | NA | NA | NA | NA | NA | NA |
| 121 | Oman | NA | NA | NA | NA | NA | NA | NA | 3.04 | 0.00 | 3.93 | 2.78 | 4.38 | 4.12 |
| 123 | Palestine | NA | NA | NA | NA | NA | NA | NA | 4.39 | 3.83 | 2.14 | 7.78 | 4.38 | 3.82 |
| 124 | Palestinian Territories | 4.743 | 0.642 | 1.217 | 0.602 | 0.266 | 0.086 | 0.076 | NA | NA | NA | NA | NA | NA |
| 126 | Papua New Guinea | NA | NA | NA | NA | NA | NA | NA | 6.03 | 6.92 | 6.07 | 3.89 | 5.63 | 7.65 |
| 133 | Republic of the Congo | NA | NA | NA | NA | NA | NA | NA | 3.31 | 3.17 | 2.50 | 3.89 | 3.75 | 3.24 |
| 144 | Somalia | 4.982 | 0.000 | 0.712 | 0.115 | 0.674 | 0.238 | 0.282 | NA | NA | NA | NA | NA | NA |
| 147 | South Sudan | 3.254 | 0.337 | 0.608 | 0.177 | 0.112 | 0.224 | 0.106 | NA | NA | NA | NA | NA | NA |
| 151 | Suriname | NA | NA | NA | NA | NA | NA | NA | 6.98 | 9.17 | 6.43 | 6.67 | 5.00 | 7.65 |
| 159 | Timor-Leste | NA | NA | NA | NA | NA | NA | NA | 7.19 | 9.08 | 6.79 | 5.56 | 6.88 | 7.65 |
| 161 | Trinidad & Tobago | 6.192 | 1.223 | 1.492 | 0.564 | 0.575 | 0.171 | 0.019 | NA | NA | NA | NA | NA | NA |
| 162 | Trinidad and Tobago | NA | NA | NA | NA | NA | NA | NA | 7.16 | 9.58 | 7.14 | 6.11 | 5.63 | 7.35 |
| 168 | United Arab Emirates | NA | NA | NA | NA | NA | NA | NA | 2.76 | 0.00 | 3.93 | 2.22 | 5.00 | 2.65 |
De lo anterior date cuenta que, por un lado, hay paises que les falta un bloque de indicadores, y que en muchos casos los nombres están mal escritos. Podemos recuperar algunos, pero en la data original:
# cambiemos a nombres usados por otra tabla:
## en demo por happy
demo[demo$Country=="Democratic Republic of the Congo",'Country']="Congo (Kinshasa)"
demo[demo$Country=="Republic of the Congo",'Country']="Congo (Brazzaville)"
demo[demo$Country=="Trinidad and Tobago",'Country']="Trinidad & Tobago"
demo[demo$Country=="North Macedonia",'Country']="Macedonia"
## en happy por demo
happy[happy$Country=="Palestinian Territories",'Country']="Palestine"Luego de esos ajustes veamos:
HappyDemo=merge(happy,demo) # re creando HappyDemo
nrow(HappyDemo)## [1] 150En efecto se recuperaron 5 paises, asi quedará.
Evaluando data
El análisis factorial requiere que hagamos algunas observaciones previas.
- Calculemos matriz de correlación:
theData=HappyDemo[,-c(1,2,9)] # sin los Scores ni nombre de país.
# esta es:
library(polycor)
corMatrix=polycor::hetcor(theData)$correlations- Explorar correlaciones:
- Sin evaluar significancia:
library(ggcorrplot)
ggcorrplot(corMatrix)
- Evaluando significancia:
ggcorrplot(corMatrix,
p.mat = cor_pmat(corMatrix),
insig = "blank")
Si puedes ver bloques correlacionados, hay esperanza de un buen analisis factorial.
- Verificar si datos permiten factorizar:
library(psych)
psych::KMO(corMatrix) ## Kaiser-Meyer-Olkin factor adequacy
## Call: psych::KMO(r = corMatrix)
## Overall MSA = 0.87
## MSA for each item =
## GDP Social Healthy
## 0.82 0.92 0.87
## Freedom Generosity Perceptions
## 0.86 0.62 0.80
## Electoral Functioning Politicalparticipation
## 0.81 0.92 0.95
## Politicalculture Civilliberties
## 0.89 0.84- Verificar si la matriz de correlaciones es adecuada
Aqui hay dos pruebas:
- Hnula: La matriz de correlacion es una matriz identidad
cortest.bartlett(corMatrix,n=nrow(theData))$p.value>0.05## [1] FALSE- Hnula: La matriz de correlacion es una matriz singular.
library(matrixcalc)
is.singular.matrix(corMatrix)## [1] FALSE- Determinar en cuantos factores o variables latentes podríamos redimensionar la data:
fa.parallel(theData,fm = 'ML', fa = 'fa')
## Parallel analysis suggests that the number of factors = 3 and the number of components = NASe sugieren 3, veamos:
- Redimensionar a numero menor de factores
- Resultado inicial:
library(GPArotation)
resfa <- fa(theData,nfactors = 3,cor = 'mixed',rotate = "varimax",fm="minres")##
## mixed.cor is deprecated, please use mixedCor.print(resfa$loadings)##
## Loadings:
## MR1 MR3 MR2
## GDP 0.268 0.927 0.126
## Social 0.321 0.663 0.152
## Healthy 0.345 0.803 0.153
## Freedom 0.255 0.284 0.516
## Generosity 0.612
## Perceptions 0.266 0.683
## Electoral 0.935 0.179
## Functioning 0.753 0.429 0.356
## Politicalparticipation 0.696 0.341 0.148
## Politicalculture 0.520 0.304 0.507
## Civilliberties 0.919 0.317
##
## MR1 MR3 MR2
## SS loadings 3.404 2.629 1.582
## Proportion Var 0.309 0.239 0.144
## Cumulative Var 0.309 0.548 0.692- Resultado mejorado:
print(resfa$loadings,cutoff = 0.51)##
## Loadings:
## MR1 MR3 MR2
## GDP 0.927
## Social 0.663
## Healthy 0.803
## Freedom 0.516
## Generosity 0.612
## Perceptions 0.683
## Electoral 0.935
## Functioning 0.753
## Politicalparticipation 0.696
## Politicalculture 0.520
## Civilliberties 0.919
##
## MR1 MR3 MR2
## SS loadings 3.404 2.629 1.582
## Proportion Var 0.309 0.239 0.144
## Cumulative Var 0.309 0.548 0.692Cuando logramos que cada variable se vaya a un factor, tenemos una estructura simple.
- Resultado visual:
fa.diagram(resfa)
- Evaluando Resultado obtenido:
- La Raíz del error cuadrático medio corregida está cerca a cero?
resfa$crms## [1] 0.03691667- La Raíz del error cuadrático medio de aproximación es menor a 0.05?
resfa$RMSEA## RMSEA lower upper confidence
## 0.07968360 0.04056531 0.10959050 0.90000000- El índice de Tucker-Lewis es mayor a 0.9:
resfa$TLI## [1] 0.9614123- Que variables aportaron mas a los factores?
sort(resfa$communality)## Generosity Freedom Perceptions
## 0.3817131 0.4118847 0.5404301
## Social Politicalculture Politicalparticipation
## 0.5661415 0.6193067 0.6232151
## Healthy Functioning Electoral
## 0.7878967 0.8784339 0.9071160
## GDP Civilliberties
## 0.9470977 0.9517927- Que variables contribuyen a mas de un factor?
sort(resfa$complexity)## Generosity Electoral GDP
## 1.039747 1.073426 1.206257
## Civilliberties Perceptions Healthy
## 1.254248 1.310446 1.439138
## Politicalparticipation Social Functioning
## 1.558243 1.566452 2.074167
## Freedom Politicalculture
## 2.077796 2.601503- Posibles valores proyectados:
¿Qué nombres les darías?
as.data.frame(resfa$scores)## MR1 MR3 MR2
## 1 -0.44581478 -1.368481159 -1.098638556
## 2 0.48033887 0.090566181 -0.856703304
## 3 -1.00967021 0.585592494 -0.898629955
## 4 -1.01725350 -0.201269389 -0.687036544
## 5 0.73624548 0.463261056 -0.807836146
## 6 0.01584794 -0.013422423 -1.259526261
## 7 1.17998664 0.708063729 1.622400763
## 8 0.82303951 0.935375820 0.642938605
## 9 -1.50616013 0.826948485 -0.466857259
## 10 -2.01760261 1.621600785 0.247169356
## 11 0.21131821 -1.001237965 0.199771946
## 12 -1.66996548 0.917390607 0.039008363
## 13 0.79578389 0.823135593 0.752042977
## 14 0.45082762 -1.668920643 0.253673440
## 15 -0.05641176 -0.551594725 0.972541028
## 16 0.50412428 -0.397523850 -0.646284689
## 17 0.08318741 0.219130546 -1.327515737
## 18 1.06057976 -0.053434870 -0.211901993
## 19 0.92310546 0.142136942 -0.826976916
## 20 0.86850458 0.416442028 -1.244491103
## 21 0.06960676 -1.591645737 0.393753312
## 22 -1.01989359 -1.843388059 -0.236504917
## 23 -1.06874704 -0.634535152 1.193457361
## 24 -0.96478982 -0.850195652 -0.051683999
## 25 1.16412448 0.699895023 1.674473964
## 26 -0.82185506 -2.292555222 -0.715561120
## 27 -1.21459683 -1.193530366 -0.601933540
## 28 1.12549099 0.310091463 0.169110288
## 29 -2.02015893 0.930753211 0.730951312
## 30 0.92382717 0.031278951 -0.555349193
## 31 -0.99341039 -0.342515557 -0.440808189
## 32 -1.34392495 -1.731710198 -0.056671606
## 33 1.11687607 0.157292047 0.196185885
## 34 0.53162832 0.480393287 -0.921160352
## 35 0.85011409 0.764563718 -0.554786856
## 36 0.79624528 0.870222852 -0.688471459
## 37 0.94143175 0.714511451 2.121511960
## 38 0.54715397 0.139014157 -0.484753176
## 39 0.50703962 -0.003093982 -0.455381011
## 40 -1.06527183 0.235186366 -0.574312978
## 41 0.68842805 -0.254455499 -1.153543849
## 42 0.89257410 0.571923939 0.195893974
## 43 -1.11306931 -1.184726903 1.091758194
## 44 1.10985513 0.659666822 1.653651197
## 45 0.81540073 0.946585860 -0.223836118
## 46 -1.16044066 0.678685564 -0.957969601
## 47 0.07691544 -0.182138682 -0.664590958
## 48 1.00677308 0.796534944 1.072782633
## 49 0.57933687 -1.062947130 0.102809649
## 50 0.94020552 0.630579743 -1.527306759
## 51 0.39726736 -0.347429130 -0.255826679
## 52 -0.78707349 -1.305764808 -0.336231492
## 53 0.46456787 -1.864571231 0.146446067
## 54 0.55857179 -0.779649140 -0.409560785
## 55 -0.17346845 1.441868311 1.010220747
## 56 0.47287912 0.678256215 -1.060140370
## 57 1.10939209 0.852767876 1.372869501
## 58 0.88601738 -0.785000631 0.101840109
## 59 0.11795711 -0.206917896 0.927890206
## 60 -2.04168760 0.889625721 0.361186027
## 61 -0.95035625 0.548246073 -1.155387122
## 62 0.98365854 1.030709219 1.423727390
## 63 0.11396222 0.866304691 1.406808120
## 64 0.79327364 0.975072334 -0.956233635
## 65 -0.44161397 -1.077594626 0.006215688
## 66 1.04591371 -0.277244425 -0.267832175
## 67 0.76222476 1.001900825 0.153818849
## 68 -0.87467550 0.112047320 0.109949230
## 69 -1.65050509 1.197143904 -0.274185318
## 70 -0.33522854 -0.966870000 1.027485617
## 71 -1.49516709 1.803872673 -0.233903317
## 72 0.04293901 -0.757298467 -0.217984352
## 73 -1.78599653 -0.152032970 0.993701862
## 74 0.97925834 0.456397863 -0.826556978
## 75 -0.77804159 0.538112456 -0.078162481
## 76 0.94805064 -1.522705964 -0.195547331
## 77 0.63831210 -2.262436649 -0.239690918
## 78 -1.68409623 0.712425727 -0.388757551
## 79 1.04840427 0.693005613 -1.315279845
## 80 0.88865588 1.352624510 1.111733995
## 81 0.34657638 0.238754003 -0.733574453
## 82 0.23975441 -1.637992753 -0.252933264
## 83 0.50924510 -2.077682623 0.466171447
## 84 0.15039097 0.505948071 0.394348464
## 85 0.63440877 -1.617044211 -0.370645384
## 86 0.80740644 0.684542762 1.164305786
## 87 -0.34912495 -0.757095434 -0.595649455
## 88 1.11202349 0.179626701 0.506248330
## 89 0.27914706 0.459796315 -0.816176737
## 90 0.69514799 -0.572741827 -0.979962610
## 91 0.71917670 -0.125566004 -0.508518883
## 92 0.21186259 0.422573984 -0.813437625
## 93 -0.45635583 -0.087937715 -0.294497203
## 94 -0.48561718 -1.783217143 0.597131853
## 95 -1.14335379 -0.633287427 1.952906573
## 96 0.45215981 -0.099779520 -0.482248625
## 97 0.08717409 -1.151560004 0.852537682
## 98 0.92601217 0.806327162 1.601247115
## 99 1.26762633 0.471511002 2.002009591
## 100 -0.86360371 -0.173889833 0.193001205
## 101 0.06111228 -2.017396366 -0.389845399
## 102 -0.19638111 -0.770587015 -0.303260964
## 103 1.02148715 1.006708243 1.940850302
## 104 -0.03154804 -0.807150396 -0.230102465
## 105 -0.61081426 -0.232953321 -0.754153058
## 106 0.73062487 0.520443632 -0.681617305
## 107 0.73511658 -0.235024164 -0.469515639
## 108 0.70482067 0.052501923 -0.874432123
## 109 0.72297250 -0.435483094 -0.288170577
## 110 0.58322251 0.726880192 -0.870804685
## 111 0.99402547 0.733652385 -0.644937904
## 112 -2.01905922 2.325144966 0.630535581
## 113 0.60145626 0.544545448 -1.211233623
## 114 -1.42279012 1.248943430 -1.411789904
## 115 -0.90602330 -1.468904392 2.288749138
## 116 -2.34549497 1.837085556 -0.350236398
## 117 0.72638640 -1.398205292 0.210006801
## 118 0.63755954 0.175566272 -0.960254960
## 119 0.29100869 -2.006041810 -0.228120837
## 120 -0.34592659 1.610737584 1.856898063
## 121 0.77684080 0.735887414 -0.978620151
## 122 0.71862466 0.806359213 -0.351334507
## 123 0.74986983 -0.082771999 -0.229027439
## 124 0.81387127 0.840110174 -0.338026089
## 125 0.86650955 0.917892294 -0.285121513
## 126 0.22068499 -0.020427117 0.300412916
## 127 -1.62467476 -0.371226704 -0.280898781
## 128 0.96997873 0.743436655 2.178871590
## 129 0.83345428 0.976979664 1.818733107
## 130 -2.13333912 -0.226400585 0.188606775
## 131 0.97446367 1.032129797 -0.543468218
## 132 -1.78808895 -0.477977648 0.094151298
## 133 0.07830330 -1.310118791 0.599597342
## 134 -0.56502405 0.585696732 0.289474776
## 135 -0.72763414 -1.559224420 -0.121188819
## 136 0.55815746 0.651184758 -0.440521935
## 137 0.09861209 0.105761486 -0.500902501
## 138 -1.22517536 0.961002640 -0.129505220
## 139 -2.31866025 0.943419620 -0.265098142
## 140 0.27386398 -1.633603584 0.420051497
## 141 0.20270976 -0.139502857 -0.874216960
## 142 1.00880031 0.718322029 0.184231055
## 143 0.57524211 1.092845257 0.369168539
## 144 1.29100183 0.219300594 0.406160014
## 145 -2.06860237 0.048333426 1.589742380
## 146 -1.20739186 0.817518450 -1.210069407
## 147 -1.55297428 0.174837293 0.656505739
## 148 -1.75630380 -0.618763549 -0.421561741
## 149 0.34238324 -1.098811528 0.358161254
## 150 -1.01759077 -1.082303889 0.303390169HappyDemoFA=cbind(HappyDemo[1],as.data.frame(resfa$scores))
library(plotly)
plot_ly(data=HappyDemoFA, x = ~MR1, y = ~MR2, z = ~MR3, text=~Country) %>%
add_markers() %>%
layout(scene = list(xaxis = list(title = 'Demo'),
yaxis = list(title = 'Tranquilidad'),
zaxis = list(title = 'Bienestar')))RECORDANDO:
library(fpc)
library(cluster)
library(dbscan)
# YA NO NECESITAS CMD para HappyDemoFA[,c(2:4)]
g.dist.cmd = daisy(HappyDemoFA[,c(2:4)], metric = 'euclidean')
kNNdistplot(g.dist.cmd, k=3)
Para tener una idea de cada quien:
resDB=fpc::dbscan(g.dist.cmd, eps=0.6, MinPts=3,method = 'dist')
HappyDemoFA$clustDB=as.factor(resDB$cluster)
aggregate(cbind(MR1, MR2,MR3) # dependientes
~ clustDB, # nivel
data = HappyDemoFA, # data
max) # operacion## clustDB MR1 MR2 MR3
## 1 0 0.2206850 2.2887491 2.3251450
## 2 1 1.2910018 2.1788716 1.3526245
## 3 2 -0.4563558 -0.2944972 1.2489434
## 4 3 -1.2251754 0.7309513 1.1971439
## 5 4 -0.5650241 0.2894748 0.5856967
## 6 5 -1.6246748 0.1886068 -0.2264006plot_ly(data=HappyDemoFA, x = ~MR1, y = ~MR2, z = ~MR3, text=~Country, color = ~clustDB) %>%
add_markers() %>%
layout(scene = list(xaxis = list(title = 'Demo'),
yaxis = list(title = 'Tranquilidad'),
zaxis = list(title = 'Bienestar')))Aqui acaba la Unidad I, el analisis factorial confirmatorio se verá en la siguiente Unidad.